A double-sided Dynamic Programming approach to the minimum time problem and its numerical approximation

Titelangaben

Grüne, Lars ; Le, Thuy Thi Thien:
A double-sided Dynamic Programming approach to the minimum time problem and its numerical approximation.
In: Applied Numerical Mathematics. Bd. 121 (2017) . - S. 68-81.
ISSN 1873-5460
DOI: https://doi.org/10.1016/j.apnum.2017.06.008

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Abstract

We introduce a new formulation of the minimum time problem in which we employ the signed minimum time function positive outside of the target, negative in its interior and zero on its boundary. Under some standard assumptions, we prove the so called Bridge Dynamic Programming Principle (BDPP) which is a relation between the value functions defined on the complement of the target and in its interior. Then owing to BDPP, we obtain the error estimates of a semi-Lagrangian discretization of the resulting Hamilton-Jacobi-Bellman equation. In the end, we provide numerical tests and error comparisons which show that the new approach can lead to significantly reduced numerical errors.